{
  "id": "2407.07756",
  "version": "v1",
  "published": "2024-07-10T15:32:38.000Z",
  "updated": "2024-07-10T15:32:38.000Z",
  "title": "Essential Semigroups and Branching Rules",
  "authors": [
    "Andrei Gornitskii"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathfrak{g}$ be a semisimple complex Lie algebra of finite dimension and $\\mathfrak{h}$ be a semisimple subalgebra. We present an approach to find the branching rules for the pair $\\mathfrak{g}\\supset\\mathfrak{h}$. According to an idea of Zhelobenko the information on restriction to $\\mathfrak{h}$ of all irreducible representations of $\\mathfrak{g}$ is contained in one associative algebra, which we call the \\emph{branching algebra}. We use an \\emph{essential semigroup} $\\Sigma$, which parametrizes some bases in every finite-dimensional irreducible representations of $\\mathfrak{g}$, and describe the branching rules for $\\mathfrak{g}\\supset\\mathfrak{h}$ in terms of a certain subsemigroup $\\Sigma'$ of $\\Sigma$. If $\\Sigma'$ is finitely generated, then the semigroup algebra corresponding to $\\Sigma'$ is a toric degeneration of the branching algebra. We propose the algorithm to find a description of $\\Sigma'$ in this case. We give examples by deriving the branching rules for $A_n\\supset A_{n-1}$, $B_n\\supset D_n$, $G_2\\supset A_2$, $B_3\\supset G_2$, and $F_4\\supset B_4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-10T15:32:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "branching rules",
      "essential semigroups",
      "semisimple complex lie algebra",
      "finite dimension",
      "semisimple subalgebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}